Prospects for toponium formation at the LHC in the single-lepton mode

This paper investigates toponium formation in the single-lepton channel at the LHC using a novel framework that integrates non-perturbative effects into Monte Carlo simulations, demonstrating that a statistically significant signal could be accessible in Run 2 data and establishing this channel as a competitive avenue for future exploration.

The Gist

The Big Idea: Catching the "Ghost" of a Top Quark

Imagine the Top Quark as the heaviest, fastest, and most impatient particle in the universe. It's so unstable that it dies almost instantly after being created—faster than it can even "hold hands" with its partner, the anti-top quark.

Usually, when physicists smash protons together at the Large Hadron Collider (LHC), they see these top quarks fly apart immediately. But, according to the laws of physics (specifically Quantum Chromodynamics), there's a tiny chance that just before they die, they might briefly form a "ghostly" bond. They swirl around each other like a pair of dancers spinning so fast they blur into a single shape before vanishing. This fleeting dance is called Toponium.

The problem? This dance happens so fast and is so subtle that it's incredibly hard to spot in the chaos of the collision data.

The New Strategy: A Different Camera Angle

For a long time, scientists tried to find this "Toponium dance" by looking at collisions where both top quarks decay into electrons or muons (particles like heavy electrons). It's like trying to film a dance in a dark room using only two cameras. It's clean, but you don't get many pictures (low statistics).

This paper proposes a new strategy: Look at the "Single-Lepton" mode.

• The Old Way (Dilepton): Both dancers fall apart into light particles. Hard to catch, but very clean.
• The New Way (Single-Lepton): One dancer falls apart into a light particle (a lepton), and the other falls apart into a messy spray of other particles (jets).

Why is this better?
Think of it like a crime scene investigation.

• The Dilepton mode is like finding two perfect fingerprints. It's great evidence, but you only find them rarely.
• The Single-Lepton mode is like finding one perfect fingerprint and a whole trail of footprints. You have way more evidence (more data), and even though the footprints are messy, you can still reconstruct the whole story if you know how to look.

The Secret Weapon: The "Re-Weighting" Trick

The authors didn't just look at the data; they built a new tool to simulate it.

Imagine you have a standard video game engine that simulates how top quarks behave. Usually, the game treats them as two separate, lonely particles. But the authors realized the game was missing a crucial rule: the "magnetic" pull that tries to bind them together right before they die.

They wrote a patch (a software update) for the game engine.

1. They took the standard simulation.
2. They added a mathematical "filter" (based on something called a Green's Function) that acts like a spotlight.
3. This spotlight highlights the specific moments where the top quarks are close enough to feel that binding force, effectively "re-weighting" the simulation to make the Toponium dance visible.

The Detective Work: Finding the Clues

Once they had this new simulation, they asked: "If Toponium exists, what does the 'footprint' look like in the Single-Lepton mode?"

They found two main clues:

1. The "Close Huddle" (Angular Correlation)
Because the Toponium dancers are spinning tightly together, their debris tends to fly out in similar directions.

• The Clue: The electron (the "lepton") and the spray of particles from the other top quark will be very close to each other in the detector.
• The Result: In the "Toponium" simulation, the electron and the jets huddle together. In the "normal" background noise, they are scattered everywhere. By cutting out the scattered events and keeping only the "huddled" ones, they found a huge signal.

2. The "Recoil Speed" (Momentum $p^*$)
This is the most exciting clue. In the world of heavy atoms, electrons orbit at a specific speed determined by the atom's size. The authors predicted that Top quarks in this "Toponium" state would have a very specific "recoil speed" (momentum) of about 20 GeV.

• The Clue: If you measure how fast the top quarks are moving in their own rest frame, the Toponium events will form a sharp peak at 20 GeV. The normal background events will just be a flat, boring line.
• The Result: Even after the messy "parton showering" (the spray of particles) and the reconstruction of the event, that sharp peak at 20 GeV survived! It's like hearing a specific musical note clearly even while a band is playing loudly in the background.

The Verdict: It's Already There!

The authors ran the numbers using data from the LHC's "Run 2" (the data collected between 2015 and 2018).

• The Result: They found that if you apply their new filters (looking for the "huddle" and the "20 GeV speed"), the signal for Toponium is statistically significant.
• What this means: They didn't just say, "We might find it someday." They said, "We could have found it already with the data we have sitting on our hard drives right now."

Summary in a Nutshell

1. The Mystery: Top quarks might briefly form a bound state called Toponium before dying, but it's hard to see.
2. The Method: Instead of looking at the rare "clean" events, the authors developed a new way to simulate and look at the "messy but common" events where one top decays into a lepton and the other into jets.
3. The Tool: They updated their computer simulations to include the "glue" that binds the quarks, making the signal pop out.
4. The Discovery: They found a "smoking gun" signature: a specific speed (20 GeV) and a specific closeness of particles that clearly separates Toponium from normal background noise.
5. The Future: This suggests that the LHC might already have the evidence needed to confirm the existence of Toponium, opening a new window into how the strong force works at the smallest scales.

The Analogy: It's like realizing that while everyone was looking for a specific rare bird in a forest by only checking the treetops (dilepton mode), these scientists realized the bird also leaves a very distinct set of footprints in the mud (single-lepton mode). By checking the mud with a new magnifying glass (the re-weighting simulation), they realized the footprints were right there, waiting to be found.

Technical Summary

1. Problem Statement

The top quark's extremely short lifetime ($\Gamma_t \approx 1.5$ GeV) prevents the formation of narrow, stable toponium ($t\bar{t}$) bound states. However, Non-Relativistic QCD (NRQCD) predicts that remnant bound-state effects (Coulombic interactions) should leave measurable imprints on the $t\bar{t}$ production cross-section near the kinematic threshold.

While recent LHC data (Run 2) has shown excesses in the dileptonic channel consistent with toponium formation, the single-leptonic channel ($t\bar{t} \to \ell \nu b q \bar{q}' b$) remains largely unexplored for this purpose. The primary challenges are:

• Theoretical Gap: Conventional Monte Carlo (MC) event generators lack a consistent treatment of non-perturbative bound-state effects in the single-leptonic mode.
• Phenomenological Need: A robust framework is required to simulate toponium formation in this channel to determine if existing LHC data can reveal these effects and to identify optimal observables for signal characterization.

2. Methodology

The authors extended their previously proposed framework (originally for dileptonic modes) to the single-leptonic channel using the MG5_AMC event generator (v3.5.7) and PYTHIA 8 for parton showering.

A. Framework Implementation

1. Process Definition: The study focuses on $gg \to (t\bar{t})_1 \to b \ell^+ \nu_\ell \bar{b} q \bar{q}'$ and its charge conjugate. Only diagrams with two resonant top propagators are included; single-resonant and non-resonant contributions are treated as "background."
2. Green's Function Re-weighting:
   • The standard Leading Order (LO) matrix elements $|\mathcal{M}|^2$ are re-weighted by the ratio of the interacting Green's function ($\tilde{G}$) to the free Green's function ($\tilde{G}_0$) of the NRQCD Hamiltonian:
     $$|\mathcal{M}|^2 \to \left| \mathcal{M} \frac{\tilde{G}(E; p^*)}{\tilde{G}_0(E; p^*)} \right|^2$$
   • Here, $E$ is the binding energy and $p^*$ is the recoil momentum of the top quark in the toponium rest frame. This factor resums Coulombic QCD interactions.
3. Code Modifications:
   • Color Structure: The intermediate $t\bar{t}$ pair is restricted to a color-singlet state. The authors modified the FORTRAN source files (matrix1_orig.f and driver.f) generated by MG5_AMC to enforce this.
   • Color Factor Adjustment: A specific modification was made to the color data arrays to account for the color structure of the $W$ decay in the single-lepton mode (multiplying by a global factor of 3).
   • Helicity Recycling: Disabled (hel_recycling = False) to prevent the generator from overriding the manual re-weighting.
4. Parton Showering Constraints:
   • To prevent the bound state from radiating gluons (which would destroy the bound-state kinematics), the authors introduced a "phantom" intermediate $Z'$ resonance carrying the toponium four-momentum.
   • This ensures PYTHIA 8 preserves the $t\bar{t}$ invariant mass and forbids radiation from the bound system, allowing only Initial State Radiation (ISR) and radiation from the final state light quarks.
   • Intermediate top quarks are explicitly re-introduced into the event record even if they are far off-shell, ensuring the full decay chain is preserved.

B. Simulation Setup

• Collision: $pp$ at $\sqrt{s} = 13$ TeV.
• Dataset: 140 fb$^{-1}$ (Run 2 equivalent).
• Parameters: $m_t = 173$ GeV, $\Gamma_t = 1.49$ GeV.
• Normalization:
  • Background: Normalized to NNLO+NNLL cross-section (243 pb).
  • Signal: Normalized to the MC-derived cross-section with re-weighting (1.51 pb).
• Reconstruction: Events were reconstructed using MADANALYSIS 5 with the anti-$k_T$ algorithm ($R=0.4$). A kinematic fit was performed to reconstruct the neutrino longitudinal momentum and assign $b$-jets.

3. Key Contributions

1. First MC Implementation for Single-Lepton Mode: The paper provides the first practical, state-of-the-art Monte Carlo simulation framework for toponium formation in the single-leptonic channel, bridging the gap between NRQCD theory and collider phenomenology.
2. Observable Identification: The authors identified specific kinematic variables highly sensitive to bound-state dynamics:
   • Angular Correlation ($min \ R^2_{\ell j}$): The squared angular distance in the transverse plane between the lepton and the nearest light jet from the hadronic top.
   • Recoil Momentum ($p^*$): The magnitude of the top quark momentum in the $t\bar{t}$ rest frame.
3. Code Availability: The authors released a Python script (reprocess_1l.py) to automate the modification of LHE files for parton showering, making the framework accessible to the community.

4. Results

The analysis applied a fiducial selection (1 lepton, 2 $b$-jets, 2 light jets, $E_T^{miss} > 30$ GeV) and a kinematic cut on the reconstructed $t\bar{t}$ mass ($m_{t\bar{t}} < 350$ GeV).

• Angular Correlation: The signal (toponium) exhibits a strong peak at small separations between the lepton and hadronic jets due to non-relativistic kinematics, whereas the background is flatter.
  • Applying a cut of $min \ R^2_{\ell j} \leq 0.5$ yielded:
    • Signal events ($N_{toponium}$): 3,060
    • Background events ($N_{t\bar{t}}$): 81,600
    • Statistical Significance ($s$): $\approx 10.5$
    • Signal-to-Background Ratio: 3.75%
• Recoil Momentum ($p^*$):
  • The toponium signal distribution peaks sharply at $p^* \approx 20$ GeV, corresponding to the inverse Bohr radius of the top quark.
  • The background distribution rises monotonically toward higher momenta.
  • Crucially, this peak structure survives parton showering and full event reconstruction, making $p^*$ a robust observable.
• Feasibility: The results suggest that a statistically significant excess ($s \sim 11$) could already be observed using existing LHC Run 2 data, assuming perfect reconstruction and no systematic uncertainties.

5. Significance

• Complementary Channel: The single-leptonic channel offers a competitive alternative to the dileptonic mode, benefiting from a larger branching ratio and simpler event reconstruction (only one neutrino).
• Direct Probe of QCD Dynamics: The persistence of the $p^* \approx 20$ GeV peak after full detector simulation provides a direct experimental handle on non-perturbative QCD bound-state dynamics at the electroweak scale.
• Discovery Potential: The study demonstrates that toponium formation is not just a theoretical curiosity but a potentially observable phenomenon with current datasets. It establishes a roadmap for future analyses to characterize and potentially discover toponium by combining these kinematic observables with other complementary variables.